LETALLEC Numerical solution of partial differential equation problems in nonlinear mechanics by quadratic minimization methods Astérisque, tome p

Astérisque

R. G

P. L

T

Numerical solution of partial differential equation problems in nonlinear mechanics by quadratic minimization methods

Astérisque, tome 132 (1985), p. 129-165

<http://www.numdam.org/item?id=AST_1985__132__129_0>

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

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R. GLOWINSKI and P. LE TALLEC Content

1. Introduction - Synopsis

2. Least squares solution of a nonlinear model problem

3. Application to the solution of the Navier-Stokes equations for incompressible viscous fluids

4. Decomposition methods by augmented lagrangians 5. Application to Finite Elasticity

References

1.Introduction. Synopsis

The solution of the Nonlinear Partial Differential Equation Pro- plems arising from Physics and Continuum Mechanics has always been an important source of challenging problems for both pure and applied mathematicians. Since the second world war the numerical solution of these problems has motivated the design of computers oriented to scien­

tific applications. Despite the impressing progresses of these compu­

ting machines (measured in terms of speed, memory, programming faci­

lities, reliability, size, cost,...) the advanced applied problems to De solved numerically have always required to work very close to the limit of the possibilities of these computers (and in fact beyond).

This situation has motivated an important effort for developping ef­

ficient numerical methods for solving the above applied problems. In this direction, an important concept is the concept of decomposition, the general principle of decomposition methods being to split the original problem in problems of smaller size and/or easier to solve, and then coordinate the local results. The coordination can be done via a least squares fitting (see Sees. 1 and 2) , or by Lagrange mul- tipliers (see Sees. 4 and 5). Domain decomposition methods (such as the Schwarz alternating method), or Alternating Direction methods founded on operator splitting provide other examples of decomposition methods. For very complicated problems, we may have to use combina­

tions of the above methods.

The main goal of this paper is to describe the numerical treat­

ment of large highly nonlinear two or three dimensional boundary value problems (originating from Nonlinear Mechanics), by quadratic minimiza­

tion techniques. These techniques have been applied to the solution of problems of practical interest and their principles are discussed in

[6] and [7] . In all the different situations where these techniques have been applied, the methodology remains the same and is organized as follows :

(i) Derive a variational formulation of the original boundary value problem, and approximate it by Galerkin methods ;

(ii)Transform this variational formulation into a quadratic minimiza­

tion problem (least squares methods) or into a sequence of quadra­

tic minimization problems (augmented lagrangian decomposition) ; (iii)Solve each quadratic minimization problem by either a direct me­

thod or a conjugate gradient algorithm with preconditioning, the preconditioning matrix being sparse, positive definite, and fixed once and for all in the iterative process.

In this paper we will illustrate the above methodology by the nu­

merical treatment of two classes of nonlinear problems : Firstly, the description of least squares solution methods and their application to the solution of the unsteady Navier-Stokes equations for incompressi- le viscous fluids, secondly the description of augmented lagrangian decomposition techniques and their applications to the solution of equilibrium problems in finite elasticity.

2. Least squares solution of a nonlinear model problem

In order to introduce the techniques which lead to the solution of nonlinear boundary value problems by least squares and conjugate gradient methods, we shall consider, first, the solution of a simple nonlinear Dirichlet problem. In section 3, these methods will then be applied to the solution of the unsteady Navier-Stokes equations for incompressible viscous fluids.

2.1. Formulation of the model problem

Let Çl c ]R^N be a bounded domain with a smooth boundary r = dQ ; let T be a nonlinear operator from V = H^(fi) to V* = H ¹(fi) , V* being the topological dual space of V. Standard notations are used for

Sobolev spaces (cf. C19J) ; in particular, (ft) denotes the space of those real valued functions defined over ft, square-integrable, with squa- re integrable first order derivatives and zero trace on r .

The nonlinear Dirichlet model problem is then (2.1)

Find u £ Hq(^) such that -Au-T(u) = 0 in H"¹(ft) .

We observe that u e H^(ft) implies that the trace of u on r vanishes ; (2.1) is therefore a Dirichlet problem. We do not discuss here the existence and uniqueness properties of the solution of (2.1) since we do not want to be very specific about operator T.

2.2. H"¹ least squares formulation of the model problem (2.1)

Many least squares formulation of the above model problem can be proposed. Among them, a natural one, based on the norm which appears naturally in (2.1), consists to say that the solutions of (2.1) cancel the norm of Au+T(u) in H ¹(ft) , and therefore minimize this norm over H¿(ft) The least squares formulation of (2.1) is then

(2.2) Min

V eEUÇl) Il Av + T(v) || -1 '

where the H (ft)-norm, || . || _^, is defined by duality, i.e.

|| f || -1 = sup V eH^ft)

<f,v> , llvl H¿(ft) = 1

where <.,.> denotes the duality pairing between H (ft) and (ft) , such that

<f,v> fv dx Vf e IT (ft) , Vv e H* (fì) .

Since the Laplace operator A is an isometry between H^(ft) and Hft ¹(ft) we can reformulate (2.2) as follows :

(2.3) Min v euhn)

Il A ¹ (Av + T(v) ) || 2 Hjjj (fi)

Taking into account the definition of the Hq(fi)-norm we introduce the function 5(v) , defined from v by

(2.4) AC = Av + T(v) in H (fi), Ç e HI(fi) ,

with £ = ? (v) . Thus £( = £(v)) is obtained from v, via the solution of a linear Dirichlet problem whose variational formulation (equiva- lent to (2.4)) is given by

(2.5) Ç e H^(fi) , fi

V£.Vw dx = fi

Vv.Vw dx-<T(v),w> Vw e H^(fi).

Using £(v) the minimization problem (2.2) can be written also as follows :

(2.6) Min

V £ H (fi) ] 2

fi

IVÇ(v)|²dx}

with 5(v) solution of (2.4), (2.5). Using (2.6) we have obtained a minimization formulation of the initial problem (2.1) ; combining (2.6) to the conjugate gradient algorithm described in Sec. 2.3, below, the solution of (2.1) will be reduced to that of a sequence of linear pro- blems associated to the Laplace operator A .

We observe that (2.6) has the structure of an optimal control problem (see [20] ) where v is the control vector, £ the state vector, (2.5) the state equation, and where the functional J : H^(fi) •> IR defined by

J(v) 1 2 fi

IV£(v)\^Zdx is the cost function.

2.3. Conjugate gradient solution of the least squares problem (2.6) . Problem (2.6) is a minimization problem. For its solution we shall use a conjugate gradient algorithm. Due to its good performances,

(cf. [22 3, [23j) , we have selected the Polak-Ribiere version of the con- jugate gradient method, that is :

Step 0 : Initialization

(2.7) u° £ H*(ft) given,

compute then g° e (ft) as the solution of (2.8) -Ag°=J'(u°) in H^_1(ft), and set

(2.9) z° = g° . •

m, j- ^ * . n n n , , n+1 n+1 n+1 .

Then for n > 0, assuming u , g , z known, compute u , g , z by Step 1 : Descent

/ -> i n, \ n+1 n , n (2.10) u = u - x z . n '

where Xⁿ is the solution of the one-dimensional minimization problem (2.11)

À e IR , n

J(uⁿ- Xⁿzⁿ) < J(uⁿ~ ÀZⁿ) VA e JR . Step 2 : Construction of the new descent direction Define g^{n + 1} e H*(ft) by

(2.12) -Ag^{n + 1} = J'(u^{n + 1}) in H"¹(ft), and set

(2.13) yⁿ = | V gⁿ-^{f l}.V(g^{n + 1}-gⁿ)dx/ j |V gⁿ | ²dx , ft

/ o i /i n n+1 n+1 . n ^m

(2.14) z = g + y ^ z . • Go back to Step 1 with n = n+1.

The two non trivial steps of algorithm (2.7)-(2.14) are :

(i) The solution of the single variable minimization problem (2.11) ; the corresponding line search can be achieved by dichotomy or Fibonacci methods. Observe that each evaluation of J(v) for a given

argument v requires the solution of the linear Poisson problem (2.4), (2.5) to obtain the corresponding £ .

(ii)The calculation of g^{n + 1} from u^{n + 1} which requires the solution of two linear Poisson problems, namely (2.4), (2.5) with v = u^{n + 1}, and (2.12) .

Let us detail the calculation of g^{n + 1}. By construction of J we have

<J ' (v) ,w> A$

fi

(v).Vn(v,w)dx Vw e Hj!j(fì), with n(v,w) (=n) solution of

n e Hq(fi) , A n = Aw + T'(v).w in H (fi) . After elimination of n we obtain

<J'(v) ,w> = fi

VUv).Vwdx-<T' (v) .w,Uv)> Vv,w £H:(Q) . Thus problem (2.12) reduces to the following linear variational

(Poisson) problem (2.15)

Find g^{n + 1} e (fi) such that Vw e H*(fi) , we have fi

Vgⁿ⁺¹.Vwdx = fi

vç^{n + 1} .Vwdx-<T"(uⁿ⁺¹).w,Çⁿ⁺¹>,

where Ç^{n + 1} is the solution of (2.4), (2.5) corresponding to v = u^{n + 1}. Remark 2.1. : As stopping criterion for the conjugate gradient algori- th (2.7)-(2.l4) we shall use

J(uⁿ) < e or ||gⁿ|| HÌ(fi] < E where e is a reasonably small positive number.

Remark 2.2. : It is clear from the above observations that an efficient Poisson solver is the basic tool for solving the model problem (2.1) by our conjugate gradient algorithm. Any size limitation for this al- gorithm will come from a limitation on the Poisson solver.

Remark 2.3. : The above methodology (possibly combined to arc length continuation methods ; cf. [8] ) extends easily to the solution of

many other nonlinear boundary value problems : Von Karman equations for thin clamped plates (cf. [24] ) , transonic flow problems (cf. [8], [ ]), etc... The choice of (2.1) as a model problem was only made for clarity reasons. In the next section we shall apply this methodology to the solution of the nonlinear elliptic system

au - vAu + (u.V)u = f in ft ,

with u a vector valued function, defined a.e. on ft , with values in N

IR . Such a nonlinear system is closely related to the solution of the time dependent Navier-Stokes equations by alternating direction methods.

3. Application to the solution of the Navier-Stokes equations for in­

compressible viscous fluids.

3.1. Formulation of the time dependent Navier-Stokes equations for incompressible viscous fluids.

Let us consider a newtonian incompressible viscous fluid ; if ft and F denote the region of the flow (ft c 1R^N, N=2 or 3 in practice) and its boundary, respectively, then this flow is governed by the Navier-Stokes equations which relate velocity and pressure inside the fluid to the external sources of motion (initial velocity, motion of the boundary, external forces, etc...) ; these equations are given by

(3.1)

du

3t vAu + (u.V)u + Vp = f in Çl ,

V.u = 0 in fi (incompressibility condition).

Above, u = ^^uj _ ^ -¹ denotes the flow velocity, p the hydrostatic pres­

sure , v the viscosity of the fluid, f the density of external forces.

Moreover (u.V)u is a symbolic notation for the skewsymmetric quadratic vector term corresponding to the convection in equation (3.1), i.e.

N 3u. N

(u.V)u = I {u

J=l ^j J 1 = 1

To fully characterize the flow, initial and boundary conditions have to be imposed on u. In the case of the airfoil A of Figure 3.1, we typically have

(3.2) u(x,t) = 0 on 3A (adherence condition on the airfoil), (3.3) u(x,t) =u^oo(t) at infinity,

(3.4) u(x,0) = u (x) (initial condition).

u -s, O O ЭА

Figure 3.1.

N

For a flow in a bounded region ft of 3R , we may replace the boundary conditions (3.2), (3.3) by

(3.5) u(x,t) = g(x,t) on r .

where due to the incompressibility condition V.u = 0, the given func­

tion g must satisfy

| g.n dT = 0 (n : unit vector normal to D . r

In the above equations the main difficulties are (i) The nonlinear term (u.V)u in (3.1) ;

(ii)The incompressibility condition V.u = 0.

Using convenient alternating direction methods for the time dis­

cretization of the Navier-Stokes equations, we shall be able to decou­

ple these difficulties ; problem (3.1), (3.5), (3.4) will reduce then to a sequence of

(a) Incompressible linear problems,

(b) Compressible nonlinear problems to be solved by the least squares- conjugate gradient methods of Sec. 2, that is via the solution of a sequence of strongly elliptic linear problems.

All the resulting linear problems will be associated (via a sui­

table space discretization) to fixed matrices. Ad hoc algorithms can

then be deviced for their numerical solution, even for large size pro­

blems. At this stage multigrid methods (cf. [15]) or domain decomposi­

tion methods (cf. [ 5 ])(or a combination of both) become very attrac­

tive .

3.2. Time discretization by alternating direction methods.

For simplicity, we suppose from now on that 9, is bounded. Let At > 0 be a time discretization step. The alternating direction method that is found to be the most convenient computationaly, to discreti- ze (3.1), (3.5), (3.4) is described just below :

(3.6) Let u° = u^Q ;

then for n > 0, and starting from u¹¹, we solve successively n+1/4 n

~ ~~ o v ^A n+l/4^ⁿ n+1/4

—Km "3 ~ ⁺!^p

(3.7) / = ?n + 1 / 4 + 1 ^A?ⁿ" ⁽uⁿ-?)uⁿ in fi ,

n n+1/4 ⁿ . ⁿ V .u ^/ =0 in 0,,

n+1/4 n+1/4 u = g on r ,

n+3/4_ n+1/4

~ ^{L t}' /² * Auⁿ⁺³^⁴ + (u^{n + 3}/⁴.V)u^{n + 3}/⁴ =

( 3-^{8 )} \ fⁿ⁺³/⁴- Vp^{n + 3}/⁴ + f Au^{n + 1}/⁴ in , n+3/4 n+3/4

u = g on r , n+1 n+3/4

u -u -2 Au + Vp _n v . n+1, ⁿ n+1 At/4

q\ ^ .n+1 ⁱ v . n+3/4 , n+3/4 ^{n N} n+3/4 . ⁿ (3.9) j f + -j Au ^/ - (u ' .V)u ^/ in 0, ,

V.u^{n + 1} = 0 in fi , n+1 n+1 u = g on r

The notation f-¹ (x) and g-¹ (x) denote f(x,jAt) and g(x,jAt), respective­

ly ; u-¹ (x) is an approximation for u(x,jAt).

Remark 3.1. : Due to the symmetrization process that it involves, 2

scheme (3.7)-(3.9) has a truncation error of 0(|At| ). Although the linear step (3.7) and (3.9) are identical, suppressing one of them would not be adviseable : it would increase the truncation error with no real gain on the computational time, which is mainly devoted to the nonlinear step (3.8).

Remark 3.2. : The decomposition of the operator - vAu between the right and left hand sides of equations (3.7), (3.8), (3.9) was done in order to involve the same linear operators in each step ; this strate­

gy results in quite substantial computer core memory savings.

Remark 3.3. : We have introduced the alternating direction decomposi­

tion of (3.1) for the continuous problems, since their formalism is simpler. But, of course, the same decomposition would apply to any Galerkin approximation of the Navier-Stokes equations, obtained by finite element methods, for example (see [7, Chapter 7]) ; actually the combination of the above alternating direction methods with spec­

tral methods of approximation is under test at the moment (the cor­

responding results will be published elsewhere).

Remark 3.4. : Related operator splitting methods for the Navier-Stokes equations are discussed in [1 ].

3.3. Least squares conjugate gradient solution of the nonlinear sub- problems (3.8).

At each full step of the alternating direction method (3.7)-(3.9) we have to solve a nonlinear elliptic problem of the following type

(3.10)

au - vAu + (u.V)u = f in ft , u = g on F.

Once the Laplace operator of Sec. 2 has been replaced by the operator al - vA, applying to (3.10) the least squares methodology of Sec. 2 yields the following minimization problem

(3.11) Min

Y^{£ V}g

J(v) ,

where

(3.12) J(v) = 1

2 ft (a|y(v) \^z+ v|Vy(v) \ *} dx, and where y(v) (=y) is the only solution of

(3.13)

y e V i o

ay - vAy = av - vAv + (v.V)v - f in V* ; the space V and the set V are defined by o g ^J

V^Q= (H^ft)), V^g= {v|v €(H^X(ft))^N, v = g on D

respectively, and V^q is the topological dual space of V^Q.

Thus, the solution of subproblems (3.8), in the alternating di- rection time discretization of the Navier-Stokes equations, reduces to the solution of the minimization problem (3.11) ; such a minimization can be achieved by conjugate gradient methods and particularly by a Polak-Ribière algorithm, like in Sec. 2. Compared to algorithm (2.7)-

(2.14) , - A will have to be replaced by al-vA ; in particular we should replace the calculation of g^{n +} in (2.12) by

(3.14)

Find ç n+1 e V such that Vwo ^ o ^£ V we have ft {agⁿ⁺¹.w + vVgⁿ⁺¹.Vw} dx =

ft

r n+1 lay .w + vVy .Vw} dx +

Jft

c n+1 ^t „^x n+1, n+1 , n+1 „^x -.j iy .(w.V)u + y . (u .V)w}dx where y = y (u ) . n+1 , n+l^N

Each iteration of the conjugate gradient algorithm applied to the solu- tion of (3.11) finally requires the solution of four linear systems as- sociated to the operator al-vA , that is

(i) One for computing y^{n + 1} (= y(u^{n +} )) through (3.13)^f (ii) One for computing g^{n + 1} through (3.14),

(iii) Two to obtain the coefficients of the quartic polynomial x -> J (u¹¹- Àwⁿ) .

In practice, the solution of the one-dimensional line search problem can be done very efficiently since it is equivalent to finding the

roots of a single variable cubic polynomial whose coefficients are known. The solution of each system associated to al-vA corresponds to the solution of N independent scalar Dirichlet problems associated to the same operator. The conjugate gradient algorithm appears then to be very efficient in this situation : usually three iterations suffice

4 6

to reduce the cost function by a factor of 10 to 10 . Therefore the whole solution of problem (3.8) by the techniques discussed in the pre­

sent section is not costly, nor for its implementation, neither for its computational running time.

3.4. Solution of the guasi-Stokes problems (3.7) and (3.9).

These linear equations, which appear at each full step of the alternating direction method (3.7)-(3.9), involve two unknowns (velo­

city and pressure) and are of the following type : (3.15)

au - vAu + Vp = f in ft.

V.u = 0 in ft , u = g on r .

Many existing solvers can be used for this problem (once a suitable space approximation has been done) : they can be direct methods such as Gaussian elimination (via an LU factorization) , Cholesky factoriza­

tion if (3.15) is approximated by a Galerkin method in which the basis functions are also divergence free (exactly or approximately), proper­

ty which allows the elimination of p in (3.15) and yields a finite di­

mensional linear system whose matrix is symmetric and positive defini­

te (see [16] for more details). One can think also to other methods for solving (3.15) such as augmented lagrangian methods (closely related to the methods discussed in Sees. 4 and 5) ; decomposition methods inclu­

ding the solution of a boundary integral problem, related to the trace of p on T,can also be used (see, e.g. [7/ Chapter 7]) leading to quite efficient Stokes solvers. Actually we can solve (3.15) (or its discre­

te variants) by a conjugate gradient algorithm quite easy to implement since it reduces the solution of (3.15) to a sequence of scalar Dirich­

let problems for al-vA ; let detail this algorithm (we still suppose N

that ft is bounded in M ).

2

We introduce first the space h c l (ft) by (3.16) h = {q|q e l²(ft) ,

ft q(x)dx = 0}

and then $ : H •+ H by (3.17)

au - vAu = - Va in fi , ~q „q ^

uq ^{= 0} on F ,

(3.18) jg^q = V.u Vq e H

It is easy to check that <£ is an automorphism from ^H onto itself, which is symmetric and H-elliptic (this last property meaning the existence of 3 > 0 such that

I (Jiq)q dx > ¡3 || q|| % Vq e H) .

fi L²(fi)

We introduce now u ^e V as the solution of

^o g (3.19)

au - vAu = f in fi, ~o -o ~ '

•u = g on r

Back to (3.17) we consider the unique pair {u,p} solution of (3.17) such that p e H ; we have then

(3.20)

a(u-u^Q)- vA(u-u^q) + Vp = 0 u - u =0 on r ,

implying, from (3.15), (3.16) that 4p = V.(u-u^Q) ,

which reduces, since V.u = 0 to (3.21) </£p = -V.u .

From the properties of i/fc it is tempting to solve (3.21) (and there- fore (3.17)) by a conjugate gradient algorithm ; we first consider the abstract form of such an algorithm

Step 0 : Initialization

(3.22) p° e H is given arbitrarily (p° = 0 for example)

(3.23) g° = Jtp° + V.u^q,

(3.24) w° = g° . •

Then for n > 0, with pⁿ, gⁿ, wⁿ known, we obtain p^{n + 1}, g^{n + 1}, w^{n + 1} as follows :

Step 1 : Descent ~ e— n„n,|2^

L" (fi) I g u ²

( 3*^{2 5 ) p}n ⁼ - a n n^

(<«w ,w ) L² (fi)

, _ _ ^x n+1 n r

(3.26) p = P - Pⁿw Step 2 : New descent direction

(3.27) g = g - pⁿ /5w , l|g^{n + 1}ll ^{2 2} (3.28) v - ^l <">

|g u ² L (fi)

(3.29) w = g ^ n^W*

Do n = n+l^f go back to (3.25).

Actually since operator Jk is not known explicitely we should proceed as follows in practice :

With p° as in (3.22), compute u° e by solving (3.30)

au° -vAu° = f - Vp° in fi.

u = g on r, o „ and set

(3.31) g° = v.u°, (3.32) w° = g°.

F o r n ^ 0, uⁿ, pⁿ, g¹¹, wⁿ being known solve ax¹¹ - vA)(ⁿ = - Vwⁿ in fi, (3.33)

Xⁿ € V^Q (= (Hj(fi))^N),

compute

(3.34) Pn =

i|gⁿll \ it (ft) ft'

n n, V.x w dx and then

/-i oirx n+1 n n (3.35) p = p - P^rw ,

,-> ⁿ+l n n

(3.36) u = u -Pⁿx , (3.37) g = g - P^RV. X , (3.38) ^n ⁼

n+1 m 2 q L² (ft) l|gⁿll ^{2 2}

L^Z (ft)

(3.39) w = g + Yⁿw ,

do then n = n+1 and go back to (3.3 3) .

Remark 3.5. : The costly part of the above conjugate gradient algorithm is the solution of the Dirichlet system (3.33) ; we observe however that the N components of x^{11 c a n} ^^e computed independently (and pos­

sibly in parallel).

Remark 3.6. : If we do not suppose that p° belongs to H we shall have convergence of {uⁿ,pⁿ} to a limit {u,p} which is the unique solution of (3.17) such that

j p dx = | p°dx . ft ft

Remark 3.7. : For very large problems, when the finite element approxi­

mation of the Navier-Stokes equations involve several ten thousands of unknowns, it might be useful to split (3.13), (3.14), (3.33) into smal­

ler size problems of the same type, obtained by domain decomposition techniques (see [ 5 ] for more details).

3.5. Numerical experiments.

We illustrate the numerical techniques discussed in the above sec­

tions by presenting some results of numerical experiments where these techniques have been used to simulate several incompressible viscous

Re = 750 ; t = .0 Figure 3.2

Re = 7 50 ; t = .2 Figure 3.3

Re = 750 ; t = .4 Figure 3.4

Re = 750 ; t = .6 Figure 3.5

flows modelled by the Navier-Stokes equations.

The experiment presented here concerns an unsteady flow around and inside a nozzle at high incidence (30 degrees) and at Reynolds number 750 (the characteristic length being the distance between the nozzle walls).The finite element methods used to approximate the Navier- Stokes equations in this experiment are discussed in [ , Chapter 7].

Figures 3.2 to 3.5 represent the streamlines at t=0, t=.2, t=.4, t=.6 respectively, showing clearly the creation and motion of eddies of va­

rious scales, inside and behind the nozzle (for more results obtained by the methods in this paper see [7, Chapter 7], [ 5 ] , [13]).

4. Decomposition Methods by Augmented Lagrangians.

The main goal of this section is to give a brief account of solu­

tion methods for variational problems when some decomposition property holds ; introducing a convenient augmented lagrangian, we obtain solu­

tion methods taking full advantage of the special structure of the pro­

blem under consideration. We shall first consider in this section the solution by augmented lagrangians of a simple nonlinear model problem, before considering in Sec. 5 the application of these techniques to the solution of nonlinear three-dimensional problems in Finite Elasti­

city .

4.1. Formulation of the model problem

Let ft c IR be a bounded domain with a smooth boundary r = 3ft, 2 and consider the following model problem (with 1 < p < +«>) ^:

(4.1)

-V.(IVu|^{p 2}Vu) = f in ft.

u = 0 on r¹,

I Vu|^p~²Vu.n = g on r² ;

in (4.1) we have t^± n t² = 0, 1^ u v² = r , dr > 0,

Such problems, discussed in [14] appear for example in the study of Norton viscoplastic fluids flowing viscously in a cylindrical duct.

Problem (4.1) is actually equivalent to the following problem of the Calculus of Variations :

(4.2)

Find u e V such that J(u) < J(v) Vv e V

where

(4.3) V = {v|v eW 'Mfi) , w = 0 on r^±} , (4.4) J(v) = 1

P fi

IVvi^P dx - fi

fvdx -

F2 gvdr

Observe now that the above functional J(.) can be naturally decomposed as follows :

(4.5)

J(v) = <^(Bv) + Ê(v) With Bv = Vv and

2 ( G ) = A

p fi

|G|^P dx, £(v) = - fi

fvdx -

r2 gvdr We have therefore for (4.1), (4.2) the equivalent formulation

(4.6)

Find {u,F} £ W such that j(u,F) < j(v,G) V{v,G) £ W

where the space W and the functional j(.) are defined (with fit = (L^p(fi))^N) by

(4.7) W = {{v,G} |v £ V, G £ H, G = Bv} , (4.8) j (v,G) = #(g) + y(v) ,

respectively. Problems (4.1), (4.2) and (4.6) are indeed equivalent but (4.8) has in some sense a simpler structure than (4.1), (4.2), des- pite the fact it contains an extra variable. This is because the linear relation Bv-G = 0 can be efficiently treated by methods using simulta- neously penalty and Lagrange multipliers, via an approximate augmented lagrangian (cf. [6]).

4.2 An augmented lagrangian associated to (4.6).

Let R be a strictly positive parameter ; we define then an augmen- ted lagrangian from V x H x H* (H* = (L^p*) ; i + = 1) into ]R by

(4.9)

S£^R(v,G,y) = e?(G) + ^(v) + i II Bv-G II Q ^ +| y . (Bv-G) dx

= ì |G|^Pdx- fvdx -j gvdr + |{||Vv-G|²+ y.(Vv-G)}dx, where ||g|| ^{n 0} = ||g|| ^{9 9} VG £ (l/(fì)) .

Remark 4.1. : We have to suppose that p > 2 in (4.9) to have Vv-G e(L²(ft))² V{v,G} e V x H ; however the algorithms obtained from the above ^ ^R behave quite well for p < 2, once (4.1) has been approximated, even by the most standard finite element methods (see [6],Ll4] for more details). •

Back to (4.9) we consider the problem

(4.10) Find {{u,F} ,A} saddle point of 25«. over {V x h} x h* , or, in other words.

Find {{u,F},A}e{V x h} x h* such that (4.11) of^R(u,F,A) <2^(v,G,A) V{v,G>£ Vx H, (4.12) st^R(u,F,A) >S^(u,F,u) Vy e H* .

Problems (4.10) and (4.6) are equivalent. Indeed, let {{u,F},X}

be a solution of (4.10). From (4.12), we have F = Vu, necessarily which means that {u,F} e W. Then if we write (4.11) with {v,G} in W we obtain

j(u,F) < j(v,G) V{v,G} £ W.

which means precisely that {u,F} is a solution of (4.6).

Conversely, if {u,F} is a solution of (4.6), by denoting

x = (F) (= |F |^P~²F) , one checks easily that {{u,F},A} is a solution o (4.10) . Observe that in this proof, the penalty term ||Bv-G|| ² ^ plays no role. But its role is fundamental in accelerating the convex gence of the numerical algorithms used for the solution of the saddle point problem (4.10).

4.3. An Uzawa algorithm for solving (4.10).

In Sec. 4.2 we have replaced the original model problem (4.1) by the equivalent saddle-point formulation (4.10). A basic algorithm for the solution of this last problem combines an Uzawa algorithm for the solution of the saddle-point problem and a block relaxation algorithm for the solution of the minimization subproblems associated to the pr mal variable {v,G} . This leads to the following algorithm

(4.13) {A⁰,u^-1} e H*x V is given ;

then for n > 0, u^{n 1} and \ⁿ being known, we compute {uⁿ,Fⁿ} by block relaxation, i.e. by setting

/ a -. a \ n n-1 (4.14) u^q = u ,

and by computing sequentially u£ and f£ by solving

(4.15) ^R(uk-l'?k'^n) " ^ r k - I ' ? ' ^ V5 £ H ; Fk e H' (4.16) ^R(uk'?k'^n) ' ^R(v,?k'~n) VV € V ;Uk £ V*

Once {uⁿ,Fⁿ} known, the Lagrange multiplier Aⁿ is updated by (4.17) A^{n + 1} = Aⁿ + p R^z(Vuⁿ-Fⁿ), p > 0,

R^z being the Riesz mapping from H onto its dual space H* (i.e.

R^z(G) = |G|^P~²G VG e H).

Remark 4.2. : Many variants of the above algorithm exists ; they are described in e.g., [6] .

Remark 4.3. : In practice, once problem (4.1) has been approximated by a convenient finite element method, we should replace R^z by the Identi­

ty mapping and take p = R, since, as shown by numerical experiments, this value of p is then quasi-optimal.

Remark 4.4. : It follows from [6] (see also [14] ) that if p > 2 (or for the finite dimensional problems approximating (4.1)) we have, for R^z replaced by 1^ and p e ]0,2R[,the following convergence results for algorithm (4.13)-(4 .17) :

lim {uⁿ,Fⁿ} = {u,Vu} strongly in V x H, lim Aⁿ = |Vu|^{p 2}Vu weakly in H*.

where u is the solution of (4.1).

Remark 4.5. : It is interesting to further analyse the structure of subproblems (4.15) and (4.16) which appear at each full step of the Uzawa algorithm used for the solution of our model problem (formulated as the equivalent saddle-point problem (4.10)). First, (4.15) in which

the variable is G, does not involve spatial derivatives of G ; if a convenient discretization of H is used, it reduces (4.15) to a family of independent algebraic pointwise problems of the following type :

(4.18) Min

G elR^{2 {}¿ |G|P ⁺ « |G-7u"H - Aⁿ.G}.

Solving problems like (4.18) is quite easy since |F^| is the solution of the one variable minimization problem

(4.19) Min ze JR

(i ^zp ⁺ | ^z2_ IrvuJ^ ⁺ xⁿ|z} .

Solving (4.19) by Newton's method is trivial ; then we obtain finally f£ from |f£| by

f£ = 0 if RVuJJ_¹ + Xⁿ = 0 , and in general

Il = & ^{( R}!^uk - i⁺ ^^{П )} l^{R v u}k - l⁺ *ⁿl

On the other hand, (4.16) is a global quadratic minimization problem given by

(4.20) Min VeV

l2 ft

|Vv-F£|²dx + ft

A .Vvdx - ft

fvdx-

r7

gvdr} ; problem (4.20) is in fact equivalent to the following linear variatio­

nal problem (4.21)

Find ufk ¹ £ V such that R 0. Vu£.Vvdx =

^ft (RF^-Xⁿ).Vvdx + ft

fvdx +

r2 gvdr Vv £V ; we observe that Vk,n problems (4.21) are associated to the bilinear

form {v,w} -M Vv.Vw dx .

Jft~ ~

Therefore we have achieved a decomposition of our model problem into a sequence of pointwise nonlinear algebraic problems which can be solved easily by Newton's type methods in IR and of elliptic linear problems, associated to the same bilinear form and whose finite dimen­

sional approximations can be solved by either direct methods (such as Cholesky's) or efficient iterative methods.

Remark 4.6. : Although not critical, the choice of good values for R is complicated ; theoretically the speed of convergence of algorithm

(4.13)- ( 4 .17) increases with R but the efficiency of the block relaxa­

tion algorithm ( 4.14)-( 4.16) deteriorates as R increases (see [6] for more details).

5. Application to Finite Elasticity 5.1. Generalities

The application of augmented lagrangian techniques to the solu­

tion of equilibrium problems in Finite Elasticity encounters three types of difficulties :

(i) The choice of reasonable constitutive laws ;

(ii) The choice of a correct functional framework for the problem de­

composition (there is no true convexity in hyper elasticity) ; (iii)The derivation of adequate iterative methods for the pointwise solution of the algebraic problems appearing after decomposition.

Since it corresponds to our more recent numerical results, we will concentrate herein on the case of compressible hyperelastic bo­

dies. The other case, concerning incompressible bodies, has already been extensively described in [ 9 ],C10] and [11 ] and is quite similar.

5.2. Formulation of equilibrium problems in compressible hyperelasti- city.

The problem that we consider consists in the determination of the final equilibrium position of an hyperelastic compressible body sub­

jected to large deformations through the application of given exter­

nal loads and imposed boundary displacements. We label any particle x of the body by its position in a stress free reference configuration

(lagrangian coordinates) and we relate both x and the displacement u(x) to a fixed cartesian system. With these conventions, the interior of the body can be identified with an open set ft of IR^N (N=2 or 3) . The body is subjected to body forces of intensity f per unit volume in the reference configuration and to surface tractions g, measured per unit area in the reference configuration, prescribed on a por­

tion T² of the boundary r of ft . Both f and g might depend on the displacement field u. This displacement field takes on prescribed va­

lues u^Q on a portion 1^ of r and we have : r = r^x v r² , r^{i n} r² = 0 .

Writting the laws of force and moment balance in lagrangian coordina- tes, and for a given law of hyperelastic type on ft , we can characte- rize formally the equilibrium positions of the considered body as the solution of the following system :

(5.1)

-V.T = f in ft , ï^Rv = g on r² (balance of forces and moments) ;

(5.2) T = — (x,I+Vu, adj(I+Vu), det(I+Vu)) (hyperelastic constitutive law) ;

(5.3) det(I+Vu) > 0 a.e. in ft , (orientation preservation) ;

(5.4) u = u^Q on Y^± .

In the above relations, T^D denotes the first Piola-Kirchoff stress tensor (whose mechanical interpretation is given in e⁴g. Ill J, C3],

£12]), v denotes the outward unit normal vector in r², before defor- mation, and W the elastic stored energy density, per unit volume of the reference configuration. Typically, this stored energy density function W is of the following type

(5.5) W(x,F,G,6) = ^(x,F) + ^(x,F,G,6),

2 where c?, is a regular convex real valued function defined on ftx ir

7 V. N² N2 , ,

and where ' from ft x ]r x ir x ]r⁺ into IR u 1+°°} can be singular at 6 = 0. For example, for OGDEN¹s type materials, we have (cf.

[ 4 1) :

(5.6) W(x,F,G,6)=C¹|F|²+ C²|G|²+ C³ 6²- C⁴ Log 6 ,

with Cⁱ, i=l,2,3,4, nonnegative coefficients, and where, for a tensor X, |x| denotes the euclidian norm of X considered as an element of

RN2.

For a constitutive law, such as (5.2), (5.5), we may formulate

the equilibrium system (5.1)-(5.4) as a variational problem as follows : Find u e {U^1,S+ u^Q} , A e (L^S (ft))^N such that

(5.7) ô = det (I+Vu) > 0 a.e. in ft.

(5.8) ft

3*

3F (x,I+Vu).Vv + A.Vv}dx = ft

f .vdx +

r2

g.vdr Vv e U '^S ,

(5.9) ft

A.F dx = $$$

3F ;x,I+Vu,adj(I+Vu),ô).F dx + 3§o

3G [x,I+Vu , adj(I+Vu) ,6)

3adj(I+Vu)

3 (Vu) ,F dx +

$$$

36 x,I+Vu , adj(I+Vu) ,6) 3 aët(I+Vu

3 (Vu) F dx VF e (L^S(ft))^N

the notation adj(F) denotes the adjugate of F, i.e. the transpose of the cofactor matrix. The usual notations are used for L^s(ft), L^S (ft)

(^s* ⁼ t~t) and the Sobolev space W^1,S(ft) ; we have

(5.10) U^{1 , s} = {y|y e (W^1,S(ft))^N, v = 0 on 1^}, (5.11) U^1,S+ u^Q = {v|v £ (W^1,S(ft))^N, v = u^q on rⁱ>.

The exponent s is related to the energy density function W by conti- nuity and coercivity properties. From a mechanical point of view tne variational problem (5.7)-(5.9) is obtained from (5.1)-(5.4) by ap- plying the virtual work principle for displacements v compatible with the boundary condition (5.4) (constraint (5.7) does not appear in the variational equations (5.8) where virtual displacement fields v pos- sibly incompatible with (5.7) are allowed ; however, for adequate W

(such as (5.6), for example), smooth solutions of (5.8), (5.9) will satisfy (5.7)). Actually the variational formulation (5.7)-(5.9) is unusual in finite elasticity, where A is often eliminated between

(5.8) and (5.9). However, formulation (b.7)-(5.9) is quite interesting since it is close from the variational formulation of equilibrium problems in incompressible hyperelasticity. Under that form, the aug- mented lagrangian decomposition described in Sec. 4, already introdu- ced in L 9 ] for incompressible hyperelasticity, can be easily genera- lized to compressible hyperelasticity.

5.3. An augmented lagrangian formulation of (5.7)-(5.9).

The extra variable needed for the decomposition of the equili­

brium equations appears quite naturally to be the deformation gradient matrix

(5.12) F = I + Vu .

The augmented lagrangian associated to our problem is then J2?^R(v,H,y) = I f^(x,I+Vv)dx + j ^ (x,H,adjH,detH) dx +

(5.13) < ^{n n}

p f 2 f

+ I |l+W-H| dx + y.(I+Vv-H)dx ,

R being an arbitrary strictly positive constant. Now, as in Sec. 4, the variational system (5.7)-(5.9) of equilibrium equations can be written equivalently as the lagrangian system below :

Find {u,F,A} e (U '^s+u )x Y x(L^S (ft)) such that

3*R I f I s

(5.14) _^(u,F,X).v = f.v dx +j^r g.v dr Vv e U^{1 , s} , g 2

3*r s N²

(5.15) -^s|(u,F,A) .H = 0 VH e (L^S(ft))^IN ,

^R f s* N²

(5.16) -^(u,F,A).y s (i+Vu-F) .y dx = 0 Vy £ (L (ft)) . In the above relations, Y denotes the set of those elements H of

(L^S(ft))^N2 such that

det H > 0 a.e. on ft .

5.4. Solution algorithm for the lagrangian system (5 .14)-(5 .16) . We apply to the solution of (5.14)- (5 .16) an algorithm, similar to (4.13)-(4.17) of Sec. 4.3, and defined as follows :

• 2

(5.17) {A°,u"¹} £ (L^S (ft))^N x {U^1,s+u^q} is given ;

then, for n > 0, u¹¹"¹ and Aⁿ being known, we compute {uⁿ,Fⁿ} by block relaxation, i.e. by

/ t- -, n \ , , . n n-1 (5.18) setting u^q = u ,

and by computing sequentially u£ and f£ by solving

83f 2 (5.19) ~af ⁽^k-i^,~k'ò^{n )}-5 ^{= 0 V}~^{H 6} (^{l S}("))^N

(5.20)

{ö (vu) ⁽^ ^⁺^ k^{) + R (}ì⁺^ k " ! k^{) +}ò^{n }}' !^{v d x =}

f.v dx + [_ g.v dT , Vv e U^{1 , S}, e U¹'^S+u

ft ^Z

and {uⁿ,Fⁿ} known, Aⁿ is updated by

(5.21) X^{n + 1} = Aⁿ + p R^z(I+Vuⁿ-Fⁿ) , p > 0 .

In the numerical applications, since we are always working with fini-

s N²

te dimensional approximations of (L (ft)) , the Riesz mapping R is replaced by the identity mapping. The updating of Aⁿ by (5.21) is then straightforward and the whole algorithm above reduces the solution of the equilibrium equations in compressible hyperelasticity to a sequen- ce of convex displacement problems (5.20) (quadratic if (5.6) holds) and local deformation gradient problems (5.19) .

Since corresponds only to the convex part of the elastic energy function W, the displacement problem (5.20) corresponds for- mally to an unconstrained uniformly convex minimization problem, set

1 s

on the linear space U ' . Among all the solution methods existing for such problems, we have chosen a conjugate gradient method with preconditioning by incomplete Cholesky factorization (icgg algorithm ; see [2l] ). The preconditioning matrix is taken symmetric, positive definite, sparse, and invariant during the iterative process. Its

"inversion" will be therefore very cheap, even in the case of large 1 s

three-dimensional finite element approximations of U ' . Due to the convexity of in (5.20), and since the solution u^_¹ at the pre- vious step is usually a good approximation of u£. The conjugate gra- dient algorithm will converge quite quickly. Nevertheless, a special attention must be paid to the choice of the preconditioning matrix, in order to avoid unnecessary oscillations in the iterative process

(5.17)- (5.21) .

5.5. Three-dimensional analysis of the deformation gradient local problems (5.19).

s N²

If (L (ft)) is approximated by a space of piecewise functions,

reduces to a sequence of independent local problems. For R sufficien­

tly large, one can prove (cf. [lb]) that each local problem is equiva­

lent to

(5.22) Min

^ ^Yloc

{^²(x,F,adjF, detF) + § | F-I-Vuj^ | ²- Xⁿ.F} , Y. = {F £ IRloc ^N , det F > 0} .

with (5.23)

If, as it is generally the case, fe' takes infinite values for non N²

positive values of det F, then we can replace Y^{1 q c} by 1R . Due to its small dimension this problem could have been solved, in

principle, by standard minimization technique for multidimensional al­

gebraic functions. But in fact, due to its local structure which in­

volves F, its adjugate and its determinant, (5.22) can be reduced to a one dimensional minimization problem if N=2, or to a sequence of one dimensional convex minimization problems if N=3. From now on we restrict ourselves to the most difficult case N=3 (three dimensional structures).

The decomposition of the local minimization problem for N=3 is again based on augmented lagrangian techniques. For that purpose, 36 9 9 we introduce three new variables f e 3R , g e JR and G e 3R which allow a simple expression of adj F and det F as functions of F and which are defined by /2 f^± =(F⁵⁺ F^g) /2 f¹³ = (F⁸⁺F³) v2 1 ^ (F^Fg)

/2 f² =(F⁵- F⁹) /2 f¹⁴ = (F⁸-F³) /2 f^- (F^Fg) /2 f³ ={F⁶+ Fg) /2 f¹⁵ = (F⁹+F²) /2 f^{2 ?}= IF³+F⁵) /2 f⁴ =(F⁶- Fg) /2 f¹⁶ = (F^y-F²) v2 ( F ^ ) /2 f⁵ =(F⁶⁺ F^?) /2 f^l7 = (F^{9 + F l}) /2 f²g= (F³⁺F⁴)

/ 2 f6 = < v ^F7> ^{/ 2 f}l8 = ⁽v^F1 > ^{/ 2 f}30= ^3"^F4⁾

/2 f⁷ =(F⁴+ F⁹) /2 f¹⁹ = (F⁷+F³) v2 (F^+Fg) /2 fg =(F⁴- F⁹) /2 f^2Q = (F⁷-F³) /2 f³²= (F.-F^

/2 f⁹ =(F⁴⁺ Fg) /2 f²¹ = (F⁷⁺F²) /2 f^{3 3}= ( F ^ )

/ 2 f10^{= ( F}4" ^F8^{} / 2 f}22 = < W ^{/ 2 f}34= ^{( F}r^F5>

/2 f^xl = (F⁵+ F^?) /2 f²³ = (Fg+F¹) /2 f^{3 5}= (F²+F⁴) /2 f^{1 2}=(F⁵- F⁷) /2 f²⁴ = (Fg-Fj) /2 f^3fi= ( F ^ )

Table 5.1.

(5.24) f = T F (T given by Table 5.1), (5.25) ^gj 2 ^£i ^r4j-4+i'

Vj=l,...,9,i=l,...,4,£¹=-£²=-£³=£⁴ = 1, (5.26) G = g .

With these new variables, it is easy to calculate adj F = G^T, det F = F.G/3

Let us now introduce the following local augmented lagrangian ((F,G},{f ,g},{z,t}) = ^U,F,G,F.G/3) + (5.27) 1

i⁺ fl^F-J-Z^uk-il² " ^ ⁺ i⁽l ! - H l^{2 +}

|g-G|² - z.(f-TF)-t.(g-G) ,

r being a strictly positive constant. The local minimization problem (5.22) can then be decomposed into the equivalent system

Find {{F,G}.{f,g},{z,t}}£ X^£ x Y^£ xz„,such that (5.28) \^,G} minimizes g?, ({.,.},{f,g},{z,t}) over X^£ , {f,g} minimizes Qf^ ({F,G},{.,.},{z,t}) over Y^£ , f = TF, g = G, (z,t> £ Z^£ ,

the sets , , being defined by 9 9 36 9 X^£ = 3ET xk", z^£ = 3R x ]p/ ,

Y£ ^{= { {}? ' ?^{} e Z}£ ' *j = 1 ^£i ^f4²j ⁺ i-4^} '

The solution of the local minimization problem (5.22) reduces, final­

ly, to the iterative solution of its augmented lagrangian formulation (5.28) by an algorithm similar to (4 .13)-(4 .17) (cf. Sec. 4.3) (with {f,G},if,g},{z,t} playing the role of u,F, A, respectively). In this algorithm, two elementary subproblems appear

(5.29) Min (F,G,f£ ,,g™ , , z^m, t^m) , {F,G}£ X^{£ 1 k}~¹ ~ ~

(5.30) Min «?. (F?,G™,f,g,z^m,t^ra) . {f,gk y^{t 1} ~^k -^k ~

Problem (5.30) is similar to (5.29), but simpler. That's why we only detail below the solution of (5.29), referring to [18] for the solu­

tion of (5.30).

5.6. Solution of (5.29).

In this section, we suppose that the stored energy function W is of OGDEN¹ s type and is given (cf.[ 4 ]) by

W(x,F,G,6) = C¹|F[² + C²IGI ² + C³ 6² - C⁴ Log 6 . For this stored energy function, the solution of (5.29) is achieved by making the following change of variables

(5.31) U = F + 3G, V = F - 3 G, with

(5.32) a = R + 4r , 3 = ((2C²+ r)/(R+4r))^{1 / 2} . Problem (5.29) is then transformed into :

(5.33) Min ^ft { ^( j U-AI ² + |V-B|²)+ C^q²-C^d Log q} , {U,V} ^€]R^{i a 4} ~ ~ ~ ~ ^{6 4}

with

(5.34) q = F.G/3 = (|U|² - |VI²)/123 ,

(5.35) A = i{R(Vu+I)-A + T^t(rf^¹-z^m) + (rg^_¹-t^m)/3} , (5.36) B = ±{R(Vu+I)-X + T^t(rf^¹-z^m)-(rg^_¹-t^m)/3} . The solution of (5.33) is easy to compute and is given by

(5.37) U = A/(f + P/63), V/(| - p/63).

where p is the unique solution in interval ]-3a$ ,3a3[ of the single variable equation

(5.38) l^|²/(f ⁺ P/63)²-|B|²/(^- p/63)²- ^(p+/p²+8C⁴C³) = 0 The numerical solution of (5.29) is thus simply obtained by

(i) Solving (5.38) by (e.g.) a Newton's method, (ii) Computing {U,V} by (5.37),

(iii) Computing {F,G} , from {U,V}, by solving (5.31), which is a trivial operation.

5.7. Axisymmetric numerical experiments.

The formulation of the equilibrium equations and the treatment of the deformation gradient local problems are given in [16J for the case of axisymmetric loadings of axisymmetric hyperelastic incompres- sible bodies. For the numerical solution of such problems, we choose

1 s here finite element approximations of the displacement space U ' and

s 5

of the deformation gradient space (L (ft)) , based on the 4 nodes asy- mmetric finite element developped by RUAS [25]. The element geometry is a triangle, the degrees of freedom for the displacements are their values at each vertex and at the midpoint of one side, the degrees of freedom for the deformation gradients are their values at the cen- ter of the triangles.

Assembling these elements 3 by 3 (as shown on Fig. 5.1) we ob- tain seven nodes symmetric finite superelements. The approximate displacements are taken continuous at element interfaces, tne approxi- mate deformation gradients are not since they are taken piecewise constant.

¿3

Al Au A²

The asymmetric finite element : Symmetric assembly of 3 elements

* degrees of freedom in deformation gradients.

• degrees of freedom in displacements.

Figure 5.1.

The numerical problem to be solved correspond then to the same lagran- gian system (5.14)-(5.16), but set on the above finite element appro-

1 s s N2

ximation of U ' and (L (ft)) . Solution techniques remain unchanged compare to those described in the continuous case. The numerical exam- ple presented in this paragraph concerns the axial compression of an axisymmetric incompressible hyperelastic shaft whose shape is indica-

ted below. For symmetry reasons, we restrict our domain ft to the up­

per meridian section of this shaft. The mesh before and after compres­

sion is represented on the figure below. We observe a surface discon­

tinuity on the shaft after deformation, such a singularity is in com­

plete agreement with the experimental studies and would be very dif­

ficult to obtain by usual numerical techniques.

Figure 5.2.

Mesh before compression

Figure 5.3.

30 % compression

Axisymmetric calculations for incompressible Mooney-Rivlin materials are discussed in Ref. [9] where comparisons with known analytical so­

lutions are also presented.

Figure 5.4.

mesh before deformation Figure 5.5.

10 % compression stable unsymmmetric solution

Figure 5.6.

Convergence rate as a function of the iteration number

5.8. Three dimensional numerical experiments.

The numerical example presented here illustrates the capability of the above numerical method for computing stable postbuckling equi­

librium positions of hyperelastic bodies, even in three-dimensional configurations. The considered body is a 2x2x20 compressible elastic beam shortened to 90% of its initial length and subjected to a very small surface pressure on one of its faces. The stored energy func­

tion of the beam is supposed to be given by the function of (5.6) (i.

e. we are dealing with an Ogden's material).

The displacement space is approximated by standard isoparametric 8 nodes hexahedral elements (Q¹ cubes), the approximate deformation gradients being constant on each element. Two solutions are then ob­

tained by using the augmented lagrangian techniques of this paragraph : (i) an unstable symmetric solution with almost no horizontal displa­

cements ;

(ii) a stable unsymmetric solution with large horizontal displacements.

The aspects of the beam before and after deformation is indicated on Figures 5.4 and 5.5, respectively. For symmetry reasons, we only con­

sider the upper part of the beam.

It is particularly interesting here to monitor the convergence of the Uzawa algorithm. Measuring the convergence rate by ||I+Vuⁿ-Fⁿ ||q ^ at each iteration, we observe that this convergence indicator first decreases while the computed solution uⁿ goes from zero to the uns­

table symmetric solution, then increases as u¹¹ automatically leaves the neighborhood of this symmetric solution, and finally decreases to­

wards zero as uⁿ approaches the final stable buckled solution (see Figure 5.6). This whole iterative process goes on completely automati­

cally without any operator's action or incremental loading technique, by the simple execution of algorithm (5.17)-(5.21) with u° = 0.

REFERENCES

[1] BEALE B.T., MAJDA A., Rates of convergence for viscous splitting of the Navier-Stokes equations, Math. Comp., 37, (1981), pp. 243- 260.

[2] BRISTEAU M.O., GLOWINSKI R., PERIAUX J., PERRIER P., PIRONNEAU O., POIRIER G., Transonic Flow Simulations by Finite Elements and Least Squares Methods, in Finite Elements in Fluids, Vol. 4, R.H. Gallagher, D.H, Norrie, J.T. Oden, O.C. Zienkiewicz eds ; Wiley, Chichester, 1982, pp. 453-482.

[3] CIARLET P.G., Topics in Mathematical Elasticity, North-Holland (to appear).

[4] CIARLET P.G., GEYMONAT G., Sur les lois de comportement en élasticité nonlinéaire compressible, C.R.A.S. Paris, 1982.

[5] DINH Q.V., GLOWINSKI R., PERIAUX J., Domain decomposition methods for nonlinear problems in Fluid Dynamics, Comp. Meth. Appl. Mech.

Eng. (to apppear).

[6] FORTIN M., GLOWINSKI R., (eds.), Méthodes de Lagrangien Augmentés^t Application à la Résolution Numérique de Problèmes aux Limites, Dunod-Bordas, Paris, 1982 (English translation published by North-Holland, Amsterdam, in 1983).

[7] GLOWINSKI R., Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New-York, 1983.

[8] GLOWINSKI R., KELLER H.B., REINHART L., Continuation-conjugate gradient methods for the least squares solution of nonlinear boundary value problems. Rapport INRIA 141, Juin 1982.

[9] GLOWINSKI R., LE TALLEC P., Numerical solution of problems in incompressible finite elasticity by augmented lagrangian methods.

(I) Two-dimensional and axisummetric problems, SIAM J. Appi. Math., 42, (1982), pp. 400-429.

[10] GLOWINSKI R. , LE TALLEC P., Numerical solutions of problems in incompressible finite elasticity by augmented lagrangian methods.

(II) Three-dimensional problems, SIAM J. Appl. Math, (to appear), [11] GLOWINSKI R., LE TALLEC P., Elasticité non linéaire : Formulation

mixte et méthode numérique associée, in Computing Methods in Applied Sciences and Engineering V, R. Glowinski and J.L. Lions eds., North-Holland, Amsterdam, 1982, pp. 281-297.

[12] GLOWINSKI R., LE TALLEC P., Augmented Lagrangians in Nonlinear Elasticity and in Plasticity (book in preparation).

[13] GLOWINSKI R., MANTEL B., PERIAUX J., Numerical solution of the time dependant Navier-Stokes equations for incompressible viscous fluids, in Numerical Methods in Aeronautical Fluid Dynamics, P.L. Roc ed., Academic Press, London, 1982, pp. 309-336.

[14] GLOWINSKI R., MARROCCO A., Sur l'approximation par éléments finis d'ordre un et la résolution par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires. Revue Française Automat.

Informat. Rech. Opérationnelle, Analyse Numérique, R-2, (1975), pp. 41.76.

[15] HACKBUSH W., TROTTENBERG U. (eds), Multigrid Methods, Lecture Notes in Math. Vol. 960, Springer-Verlag, Berlin, 1982.

[16] HECHT F., On weakly divergence free finite element bases (to appear).

[17] HUGHES T.J.R., MARSDEN J., Mathematical Foundations of Elasticity, Prentice Hall, Englewood Cliffs, N.J., 1983.

[18] LE TALLEC P., VIDRASCU M., Une étude numérique pour les problèmes d'équilibre de corps hyper élastiques compressibles en grandes déformations. Rapport Sectoriel MA 1 du Laboratoire Central des Ponts et Chaussées, 1983.

[19] LIONS J.L., Problèmes aux Limites dans les Equations aux Dérivées Partielles, Presses de l'Univers, de Montréal, Montréal, P.Q., 1962.

[20] LIONS J.L., Controle Optimal des Systèmes Gouvernés par des Equations aux Dérivées Partielles, Dunod, Paris, 1968.

[21] MEIJERINK J.A., VAN DER VORST H.A., An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comp., 31, (1977), pp. 148-162.

[22] POLAK E., Computational Methods in Optimization, Academic Press, New-York, 1971.

[23] POWELL M.J.D., Restart procedure for the conjugate gradient method. Math. Programming, 12, (1977), pp. 148-162.

[24] REINHART L., On the numerical analysis of the Von Karman equa- tions : Mixed Finite Element Approximation and Continuation Techniques, Numerish Math., 39, (1982), pp. 371-404.

[25] RUAS V., Méthodes d'éléments finis en élasticité incompressible non linéaire. Thèse d'Etat, Université Pierre et Marie Curie, Paris, 1982.

R. GLOWINSKI

University P.et M. Curie 4, place Jussieu

75230 PARIS CEDEX 05 P. LE TALLEC

L.C.P.C.

Service Mathématiques 58, Bd. Lefèbvre 75732 PARIS CEDEX 15